What is the integral of (2x-5)/((x-1)^2) ?

The integral of (2x-5)/((x-1)^2) is 2ln|x-1|+3/(x-1)+C

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integral of (2x-5)/((x-1)^2)

\int \frac{2 x - 5}{( x - 1 ) ^{2}} d x

2 ln ∣ x - 1 ∣ + \frac{3}{x - 1} + C

Solution steps

\int \frac{2 x - 5}{( x - 1 ) ^{2}} d x

Apply u-substitution

: \int \frac{2 u - 3}{u ^{2}} d u

= \int \frac{2 u - 3}{u ^{2}} d u

Expand

\frac{2 u - 3}{u ^{2}} : \frac{2}{u} - \frac{3}{u ^{2}}

= \int \frac{2}{u} - \frac{3}{u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2}{u} d u - \int \frac{3}{u ^{2}} d u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

\int \frac{3}{u ^{2}} d u = - \frac{3}{u}

= 2 ln ∣ u ∣ - (- \frac{3}{u})

Substitute back

u = x - 1

= 2 ln ∣ x - 1 ∣ - (- \frac{3}{x - 1})

Simplify

= 2 ln ∣ x - 1 ∣ + \frac{3}{x - 1}

Add a constant to the solution

= 2 ln ∣ x - 1 ∣ + \frac{3}{x - 1} + C

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What is the integral of (2x-5)/((x-1)^2) ?
The integral of (2x-5)/((x-1)^2) is 2ln|x-1|+3/(x-1)+C